One of the things that was really intriguing to me that was discussed in class was the idea of payoff matrices in real life situations. Take for instance, the decision made when people compete at a tournament for prizes. What motivates people to wholeheartedly compete for first prize? Would it be more practical to save the effort and just agree to split the prize afterwards?
One example I had in mind when I first started thinking about this was during a Magic: The Gathering draft. These are small, 8-man single elimination tournaments for a card game, for any people unfamiliar with what Magic is. Two such prize formats that exist are as such: 8 to the winner and 4 to the runner up, and 4 to the winner, 3 to the runner up, and 2 to the two semifinalists.
Just analyzing the 8-4 format, lets say that the player has a probability p of winning the match. If both players decide to split, then the payouts would be 6,6. If they choose to play, then the payoffs for the winner would be 8p + 4(1-p) where p (or q) is the probability you think you have of winning. If you had absolute confidence that you would win, splitting would definitely not be an optimal choice for you, but if that confidence is at any point below 1/2, then it would be more beneficial to split. The other side of this is that you can always lie, and “agree” to split but actually just walk away with the full prize after the other person concedes. This would result in a payoff of 8,4 if player A lies. In reality, however, there is some consequence C for being dishonest, whether that is administrative punishment, or a blow to your credibility. So in actuality, the payoff is 8-C,4.
Setting up the payoff matrix is as such (if one person splits and the other person does not, assume the person who does not is lying)
Split No Split
Split 6.6 4, 8-C
No Split 8-C,4 8p+4(p-1),8q+4(q-1)
The existence of equilibria is all dependent on what values a person sets for C, p, and q. Seeing as how dishonesty is usually very much frowned upon, then C would oftentimes be much greater than 2, or 8 for that matter. Nash equilibria would exist where both players choose to split, or both players would choose to play. It is interesting to note, however, that a dishonest person who values C at 0 would displace Split,Split as an equilibrium. In the case of dishonesty, the only equilibrium would be to play every game, or risk losing for sure.
In reality, splits happen very often. This is likely due to the nature of good policing for the system (disputes are handled promptly and offenders punished), as well as the tightness of the network that plays Magic: The Gathering online. People seem less likely to cheat those whom they identify as similar to themselves, and the smaller community lends itself to good behavior.
This analysis does not take into account competitive personalities, pride of winning, or any other intangible factors that might influence the payouts. It is interesting what insights this model gives about the community that plays Magic: The Gathering online, as well as insights into decision-making during games.